Uncertainty in complex valued quantities

When considering the uncertainty associated with a real valued measurement of a system it is very common to assume that the variable is drawn from a normal probability distribution [77]. The assumption of normality allows the distribu- tion to be defined by the statistics of the normal distribution, the mean µ and the variance σ2_{. The uncertainty can then be displayed by a percentage confidence} interval which defines the interval in which the measurement falls within a per- centage probability, see Figure 7.1A.

When the measurement is drawn from a bivariate normal distribution the statistics are defined by the vector mean µ and covariance given by

Σ= " σ1,1 σ1,2 σ_2,1 σ2,2 # (7.1)

where σ1,1, σ2,2 are the variance in the 1’st and 2’nd variate and σ1,2 = σ2,1 is the covariance between the two. The mean and covariance matrix define a probability distribution in the space to of two variates that characterise the uncertainty. Anal- ogous to the univariate measurement where the uncertainty can be displayed as a confidence interval, for the bivariate measurement a percentage confidence area is defined by the uncertainty in each variate and the correlation between them [48]. In the space of the two variates the confidence area is an ellipse.

A complex valued variable is often represented in two parts, real and imagi- nary (commonly plotted on an argand diagram), where

x=a+jb=Re(x) +j Im(x). (7.2) Complex variables can hence be though of as a bivariate measurement:

X= [Re(x), Im(x)]. (7.3)

In general the variance in the real and imaginary parts of the measurement will not be independent and can therefore be assumed to be drawn from the bivariate

normal distribution X∼ N X| " µ_Re(x) µ_Im(x) # ,ΣRe(x),Im(x) ! . (7.4)

Figure 7.1: The uncertainty in a complex variable can be represented by a bivari- ate normal distribution creating an elliptical uncertainty area in the real imag- inary space. A) The probability distribution of a real valued univariate variable with its 95% confidence intervals. B) The elliptical confidence area of a complex variable represented by a bivariate normal distribution.

The uncertainty in a complex variable is then displayed as an elliptical confi- dence area in the real-imaginary space, see Figure 7.1B. Another alternative repre- sentation for complex variables is in gain-phase form, this is very common when considering systems in the frequency domain and has been used throughout this thesis. The gain phase representation of a complex variable can also be consid- ered as bivariate and so can be treated similarly. Care should be taken when using the gain phase representation however because the uncertainty predictions at near zero gain can be inaccurate [102].

In Chapters 3 and 4, different frequency domain descriptions, based on GFRFs and NOFRFs were introduced, that extend the linear FRF to higher order non- linear systems. The FRFs of all orders are complex valued and can be assumed to be drawn from a bivariate normal distribution. The following example demon- strates how the uncertainty is manifested in the FRF of a linear system, however the concept can easily be extended to the non-linear case.

Consider the following generative ARX model yk = θ₁y_k−1+θ2y_k−2+θ3u_k−1+θ₄u_k−2+e_k (7.5) where θ=       θ1 θ2 θ3 θ4       =       0.2 0.1 0.1 0.05       . (7.6)

where ek is an i.i.d white noise sequence drawn from the normal distribution ek ∼

N (ek|0, σ22) where σe2 = 0.0005. The system is simulated for N = 1000 samples in response to the input excitation signal uk drawn from a uniform distribution in the range[₋0.5, 0.5].

Parameters are estimated using VB inference by Algorithm 6.2, initialised with a0 = c0 = 1×10−2 _{and b0 = d0 = 1×10}−4 _{. The resulting posterior distribution} on the parameters is normally distributed with mean and covariance given by

θµ=       0.1731 0.1285 0.0989 0.0520       , Σθ=       0.7909 −0.3635 −0.0010 −0.0773 −0.3635 0.4432 −0.0016 0.0347 −0.0010 −0.0016 0.0063 0.0002 −0.0773 0.0347 0.0002 0.0138       ×10−4_{. (7.7)}

The frequency domain description of the system given by Equation (7.5) is its first order FRF,

H1(ω, θ) = θ3e

−iω₊

θ4e−2iω

1−θ₁e−iω−θ2e−2iω. (7.8)

The FRF, H1(ω, θ)is therefore a function of an uncertain variable, namely the

vector θ, and therefore it is itself uncertain. Furthermore it is also known to be complex valued necessitating the use of complex uncertainty analysis.

A Monte Carlo simulation is performed on H1(ω, θ)by drawing N_MC = 1000

parameter values randomly from the multivariate normal distribution_{N (}θ|µθ,Σθ).

The real and imaginary parts of H1(ω, θ)are plotted against each other for each

Monte Carlo sample at three different frequency values, ω = 5.52, 5.64 and 5.77, see Figure 7.2. The covariance matrices between the real and imaginary parts of H1(ω, θ)are estimated from the Monte Carlo samples, assuming a bivariate nor-

mal distribution, and are given by

ΣH1(θ,ω=5.52) = "0.1193 0.0157 0.0157 0.1399 # ×10−4, ΣH1(θ,ω=5.52) = "0.1295 0.0017 0.0017 0.1698 # ×10−4,

ΣH1(θ,ω=5.52)=

"0.1169 0.0100 0.0100 0.2287 #

×10−4. (7.9)

The covariance matrices are used to calculate 90%, 95% and 99% confidence intervals which are overlaid on the sampled FRF. The real and imaginary parts of H1(ω, θ)are also plotted against each other at all frequencies for the true param-

eter vector.

Figure 7.2: The uncertainty associated with the Frequency response function can be described by the covariance between its real and imaginary parts.Scatter plots of the real vs imaginary parts of H1 calculated from Monte Carlo samples of the parameter distribution at ω = 5.52, 5.64 and 5.77 plotted with confidence

bounds and the real vs imaginary parts of H1at the true parameter vector.

The Figure shows that in this case the approximation that H1(ω, θ)can be de-

scribed by a normal distribution is a good one. The distributions are also skewed, indicating that it is necessary to consider covariances between the real and imag- inary parts. It is also clear that the uncertainty in the FRF is a function of the frequency.

4

The above example provides a motivation for the uncertainty analysis pre- sented in the remainder of this chapter. It shows that there is covariance between the real and imaginary parts in the FRF and that these depend on the uncertainty in the model parameters as well as on the frequency.